Res. Agr. Eng.
Vol. 57, 2011, No. 3: 79–87
Analysis of force effects measured in the tractor
three-point linkage
J. Čupera, F. Bauer, L. Severa, M. Tatíček
Faculty of Agronomy, Mendel University in Brno, Brno, Czech Republic
Abstract
Čupera J., Bauer F., Severa L., Tatíček M., 2011. Analysis of force effects measured in the tractor three-point
linkage. Res. Agr. Eng., 57: 79–87.
The work is focused on description of tractor individual axels and wheels loading during ploughing. A combination of
tractor inclination with force and momentum effects need not always result in marked differences in the load of driving
wheels. This paper documents the relationship existing between forces working in the upper drawbar Fh of the threepoint linkage and the momentum MTx. It was also demonstrated that with the increasing value of MTx the difference
between loads of both rear wheels reduces. The described algorithm enables to evaluate output parameters of ploughing
aggregates for different variants of forces changing in the tractor three-point linkage due to various ways of suspension
and adjustment of individual plough types. When using data recorded in the course of ploughing operations it is possible
to demonstrate the effect of plough adjustment/attachment on the efficiency of the whole ploughing aggregate.
Keywords: three-point linkage; wheel load; load cell
The development of tractors and tools directed
to system with either attached or trailed machines
resulted in certain discrepancies between the manner of operation of tractors and the methods of
measuring their tractive properties. Attached and/
or trailed sets of machines are characterised by the
fact that the tools are connected with the tractor by
means of a three-point linkage so that their resulting force effect on tractor is usually markedly different from effects of simple tractive forces, which
typically exist in each tractive test of tractors.
Measurements of forces existing in the threepoint linkage of a tractor and the projection of
measured force effects into its centre of gravity were
described by Bauer et al. (1989). Results of the first
experiment with the determination of effects of the
ploughing operation on soil compaction were presented in a paper published by Mišun and Bauer
(1992). Problems associated with force effects of
ploughs on tractors ploughs kinematics were studied by many authors (Kazzaz, Grečenko 1968;
Upadhyaya, Kemble 1985; Sloboda et al. 1997;
Mouazen et al. 2009). Also other factors affecting drawbar characteristics such as tire inflation
and others were deeply studied (Novák et al. 2009;
Šmerda, Čupera 2010).
The objective of this paper is to describe forces
influencing wheels of a ploughing tractor and to
find out which of them put a concrete load on driving wheels because the driving wheels are of crucial importance for the tractor’s tractive efficiency.
Concerning tractive efficiency, there is a concern
to reach the maximum values of this factor but, regarding effects of an undesirable compaction of soil
(and subsoil), the adhesion load of driving wheels is
in conflict with this requirement above all in case of
Supported by the Ministry of Education, Youth and Sports of the Czech Republic, Project No. MSM 6215648905 and
Internal Grant Agency TP1/2011.
79
Vol. 57, 2011, No. 3: 79–87
ploughing aggregates, with right wheels moving in
the furrow. As far as these ploughing aggregates are
concerned, many authors refer to the compaction
of subsoil due to the movement of tractor wheels
in the furrow (Upadhyaya, Kemble 1985; Renius
1987). The analysis of forces influencing the threepoint linkage of a ploughing tractor was performed
due to the fact that, as compared with tractor tractive tests, the load of a ploughing tractor shows
characteristic differences. When ploughing, the
tractor wheels move on one side in the furrow so
that it is inclined. In combination with power effects of the plough this inclination causes a different
load of individual wheels and changes in the load of
both axles. This results in significant changes not
only in the grip of driving wheels but also in an increased compaction of soil. Performed analysis can
be used for further studies on dynamic properties
of tractor units or other vehicles as described e.g. in
Chalupa et al. (2006, 2009).
Material and methods
Field measurements were performed on the farm
Agroservis Višňové (Znojmo district, Czech Republic), on a field with sandy-loamy soil (the average moisture content W h was 17 vol. % and the
measured specific resistance of plough was 69 kPa).
All measurements were performed under identical
conditions. The total area of the experimental field
Res. Agr. Eng.
was 60 ha and the measurements were performed
on a plot 50 m long. Prior to the measurements the
plough and its linkage were adjusted in such a way
that it was possible to observe effects of different
lengths of the upper bar on selected parameters of
the ploughing aggregate. During all measurements
it was strictly controlled that the plough was correctly adjusted and that the ploughing result met all
agrotechnical requirements.
This paper presents results of those measurements,
which demonstrate problems associated with the load
of driving wheels of a tractor when ploughing either
on its right or the left side with an attached reversible
plough. As during nearly all tests not only the forces in
drawbars of the three-point linkage but also the tractive force existing in the traction rope (the ploughing
tractor was pulled by another tractor by means of a
rope), the measured force Fts could be used as an input value not only for the estimation of the momentary average coefficient of the rolling resistance fv but
also of the momentary average coefficient of adhesion µ. However, this could be performed only on the
assumption that also another input value – resistive
force Fw (required for the surpassing of the resistance
in gearings of the pulled tractor) would be exactly
measured. Measurements of gear resistance were
performed on a roller dynamometer in laboratories
of Mendel University in Brno, Brno, Czech Republic.
Measurements of forces influencing the ploughing tractor were performed by means of five tensometric sensors fitted to drawbars of the three-point
Point M is freely moving in the slot opening
Fig. 1. Three-point linkage fitted with load cells
80
Res. Agr. Eng.
Vol. 57, 2011, No. 3: 79–87
Fig. 2. Schematic presentation of three-point
linkage with load cells
S – load cell; AP, AL – Linkage of lower draw
bars to the tractor; P – right; L – left; BP, BL –
linkage of lower draw bars to the plough; DP,
DL – linkage of lift-up draw bars to the tractor;
H – linkage of the upper draw bar to the tractor;
M – linkage of the upper draw to the plough;
Fpz – force in the right lift-up draw bar; Flz – force
in the left lift-up draw bar; Fpd – force in the right
lower draw bar; Fld – force in the left lower draw
bar; Fh – force in the upper draw bar; Fts – pulling
force; FALx – force in the point A situated left from
the x axis; FALy – force in the point A situated left
from the y axis; FALz – force in the point A situated
left from the z axis
linkage and also to the pulling rope (Fig. 1). When
analysing the measured values, we have calculated
average sectional values of axial forces occurring in
drawbars and the average sectional value of the force
Fts existing in the rope; these forces occurred as a
response to the ploughing, i.e. as a set of values
{Fld, Flz, Fpd, Fpz, Fh, Fts}
(1)
where:
Fld, Fpd – average sectional values of axial forces working
in the lower drawbars (kN)
Flz, Fpz – average sectional values of axial forces working
in the lifting drawbars (kN)
Fh
– average sectional values occurring in the upper
drawbar (kN)
Fts
– average value of the traction force existing in
the rope (kN)
The average sectional forces were transformed
into the gravity point of the tractor within each
time interval i = 1... n using the relationship
Af = [FTx, FTy, FTz, MTx, MTy, MTz]T
(2)
is the matrix of transformation of binding effects
from points AL, AP, DL, DP, and H (Fig. 2) into the
tractor’s centre of gravity; x, y, z are coordinates of
corresponding points AL, AP, DL, DP and H and
f = [fAL, fAP, fDL, fDP, fH]T (kN)
(4)
is the vector of force effects in the aforementioned
binding points. It is also obvious that
fAL = [FALx, FALy, FALz]T (kN)
(5)
fAP = [FAPx, FAPy, FAPz]T (kN)
(6)
fDL = [FDLx, FDLy, FDLz]T (kN)
(7)
fDP = [FDPx, FDPy, FDPz]T (kN)
(8)
fH = [FHx, FHy, FHz]T (kN)
(9)
If the product of Eq. (2) is, among others, also the
force FTx, it is obvious that the difference
Fvm = Fts – FTx = Fv + Fw
(kN)
(10)
where: 0
0
1
0
0
1
0
0
1
0
0
1
0
0 º
ª 1
« 0
1
0
0
1
0
0
1
0
0
1
0
0
1
0 »»
«
« 0
0
1
0
0
1
0
0
1
0
0
1
0
0
1 »
A «
»
zAL yAL
0
zA P y A P
0
zDL yDL
0
z Dp y Dp
0 zH yH »
« 0
« zAL
0
x A L zA P
0
xA P z D L
0
xD L z Dp
0
xDp z H
0 xH »
»
«
0
y A P xA P
0
y D L xD L
0
y Dp xDp
0
y H xH
0 »¼
¬« y A L xA L
(3)
81
Vol. 57, 2011, No. 3: 79–87
Res. Agr. Eng.
Fig. 3. Schematic presentation of forces impacting to the
tractor
P – point of force impact to the right wheel; L – point of
force impact to the left wheel; C – point of force impact
to the front axle; T – tractor’s point of gravity; FPx, FPy,
FPz – forces impacting the right wheel in axes x, y, z; FLx,
FLy, FLz – forces impacting the left wheel in axes x, y, z; FCx,
FCy, FCz – forces impacting the front axle in axes x, y, z; FTx,
FTy, FTz – forces in the point of tractor’s gravity in axes x,
y, z; MTx, MTy, MTz – momentums in the point of tractor’s
gravity around axes x, y, z; Fts – pulling force; α – inclination
of tractor; x, y, z – coordinates
–
measurement, then the arithmetic midpoint F w of
Fw values can be expressed by the equation
is equal to the sum of the Fv force of rolling resistance of wheels of the pulled tractor with the resistance force Fw of freely rotating tractor’s parts. It
results from Eq. (10) that the resistance is the force
of rolling resistance
Fv = Fts – FTx – Fw
(kN)
– –
–
–
–
F w = F ts – F Tx – f v(F Tz + Gcos α)
(kN)
(12)
and the average momentary sectional value of the
coefficient of rolling resistance
f
v
Fv
FTzc
Fts FTx Fw
FTz G cos D
(–)
(14)
The statistical analysis of measurements of
ploughing operations at the measured concrete
–
–
value f = 0.08 produced F w values ranging from
1.8–2.4 kN (Eq. 14).
When modelling the tractor’s drive, the front
wheels were replaced by a single wheel with a doubled reaction 2 × F Cz (Fig. 3) and with the total side
effect F Cy, which influences both front wheels. Due
to the tractor’s inclination such a single wheel follows an imaginary pathway but has the same properties ( f v, µ) as rear wheels.
The transformation of binding reactions and reactions of soil to the tractor’s movement (as projected into its centre of gravity) and the equation of
the balance of actions and reactions in this centre
of gravity is expressed by the equation
(11)
If α is the tilt angle of the tractor driving with
left wheels in a furrow then the total vertical force
affecting the tractor in its centre of gravity is expressed by the equation
FTzc = FTz + Gcos α
(kN)
(13)
where:
α – oriented angle of the tractor’s inclination
BfR + [FTx, FTy + Gcos α, FTz + Gcos α, MTx, MTy, MTz]T
– –
–
If F ts, F Tx and F Tz represent arithmetic midpoints
of corresponding sectional values of a concrete
= 0T6
(15)
where:
ª 1
« 0
«
« 0
B «
0
«
z
« C
« y
¬ C
82
0
1
0
0
1
0
0
1
0
zC
yC
0
xC fvr zL
0
yL
x C fvr
0
0
1
0
0
1
0
0
zL
0
x L f vR
1
yL
x L f vR
0
0
zP
yP
0
º
»
»
0
1
»
» (16)
zP
yP
»
x P f vR »
0
»
x P f vR
0
¼
0
1
0
0
Res. Agr. Eng.
Vol. 57, 2011, No. 3: 79–87
is the matrix of transformation of binding reactions
of the plough into the tractor’s centre of gravity,
where x, y, and z are coordinates of points L, P, and
C of the tractor; r and R represent radius of wheels
and
fR = [fC, fL, Fp]
(kN)
(17)
is the vector of the binding reactions of soil to the
tractor’s movement.
As the modelling of this problem was based on an
assumption that both the front double wheel and
the rear wheels are driving wheels with differential
locks, then
640 mm, 665 mm, and 695 mm, respectively. The
measured values indicate that in Group A (with
the shortest drawbar) a marked tractive force was
recorded in all measurements and that its average
value was 11.93 kN. In Group B the length of upper drawbar was 665 mm and it was found out that
due to the elongation by 25 mm the tractive force
decreased to 5.01 kN. In Group C the length of the
upper drawbar was 695 mm and the point M moved
freely within the oblong opening of the plough (see
a detail in Fig. 1). Theoretically, there should not be
any force measured in the upper drawbar. However,
the bumpiness of the field surface caused that the
average values of measured forces were not zero
but fluctuating about zero values.
fC = [–2F Cz(μ – f v), F Cy, –2F Cz]T
(kN)
(18)
fL = [–FL(μ – f v), F S, –FL]T (kN)
(19)
Table 1. Average force values measured for different
lengths of upper drawbar
fP = [–FP(μ – f v), 0, –FP]T (kN)
(20)
Measurement
series
provided that the denomination
FL = FLz, FP = FPz, F S = FLy + FPy
A
is used in accordance with Fig. 3.
Eq. (15) results in obtainment of six equations for
a set of six unknowns
Sn = {μ, F Cy, F Cz, FL, FP, F S}
(21)
Results
The algorithm described above indicates that it is
possible to calculate forces and moments working
in the tractor’s centre of gravity as well as forces
influencing its rear wheels and the forces occurring
in the point C (Fig. 3). As the objective of this paper
is to explain reactions of soil to the movement of
rear wheels of the ploughing tractor (which participate in a decisive manner in an efficient transfer of
the driving moment from wheels on the soil surface) the tables contain only those values that are
directly related to these problems.
A series of measurements presented in Table 1 refer to those ploughing operations, which were performed on the left side of the pathway of a ploughing tractor. The upper drawbar of the tractor’s
three-point linkage was adjusted to three different
lengths. For Groups A, B and C of measurements,
the lengths of the upper drawbar were adjusted to
Average
B
Average
C
Average
Fh (kN)
MTx (kNm)
FLz – FPz (kN)
11.350
–1.840
3.710
11.384
–1.918
4.038
11.692
–1.880
3.665
12.669
–1.777
3.459
12.554
–2.010
3.091
11.930
–1.885
3.592
4.041
–0.209
6.278
6.088
–0.518
5.278
3.245
–0.054
5.601
4.515
–0.013
5.739
6.018
–0.574
5.080
6.152
–0.519
5.048
5.010
–0.314
5.504
0.109
0.551
7.117
0.140
0.715
7.231
0.429
0.420
7.022
–0.113
0.324
7.024
–0.060
0.177
6.792
1.334
0.389
6.924
0.293
0.034
6.861
–0.030
0.402
7.231
Ploughing on left; A – length of upper drawbar 640 mm;
B – length of upper drawbar 665 mm; C – length of upper
drawbar 695 mm; F h – force acting in the upper drawbar
of tractor three-point linkage; MTx – momentum on axes x;
F Pz – F Lz – difference of reactions on rear wheels
83
Vol. 57, 2011, No. 3: 79–87
Res. Agr. Eng.
Table 2. Test of regression functions
Regression equation
R
F
Significance F
Coefficient
P value
Coefficient
P value
(FPc – FLc) = 4.3345+ 0.7571 MTx 0.8620 26.03463
0.000643
4.33453
3.57E-11
0.757112
0.000643
MTx = –0.65968 + 0.15458 Fh
0.000161
–0.65968
0.010044
0.15458
0.000161
0.8998 38.27281
(FLc – FPc) = 6.2863 + 1.4764 MTx 0.9711
281.085
5.23E-12
6.286305
2.7E-22
  1.476383
5.2E-12
MTx = 0.489148 – 0.19162 Fh
365.073
6.31E-13
0.489148
1.5E-06
–0.19163
6.3E-13
0.9775
Fig. 4. Dependence of differences in forces
influencing the rear wheels FLc – FPc on MTx
momentum
FLc – FPc (kN)
Ploughing on left
MTx (kNm)
the obtained results are presented in Table 2. These
results indicate that the calculated coefficients were
statistically highly significant.
Dependencies presented in Figs 4 and 5 were
calculated by means of regression analysis. The dependence of the linear course of the moment MTx
on the force working in the upper drawbar of the
three-point linkage of a tractor is presented in Fig. 4.
Increasing negative values of the moment MTx (as
related to the orientation of the x-axis indicate
MTx (kNm)
Results of these measurements were analysed statistically by means of the regression analysis. The
measured values were plotted along a straight line
using the results of linear regression. The statistical
significance of the regression function was tested
by F-test. Test results presented in Table 2 indicate
that the tested regression was statistically significant (this was also documented by the calculated
significance levels). The significance of the regression function coefficients was tested by t-test and
Fh (kN)
84
Fig. 5. Dependence of MTx momentum on
force Fh in the upper drawbar
Ploughing on left
Res. Agr. Eng.
Vol. 57, 2011, No. 3: 79–87
Table 3. Average force values measured for different
lengths of upper drawbar
Measurement
series
A
Average
B
Average
Fh (kN)
MTx (kNm)
FPz – FLz (kN)
10.708
1.418
5.459
11.068
1.140
5.451
11.417
1.298
5.265
11.694
1.262
5.307
11.222
1.279
5.370
6.330
0.098
4.469
8.555
0.174
3.822
7.047
0.100
4.117
6.774
0.073
4.341
1.317
–0.179
4.403
3.093
–0.042
4.249
1.684
–0.280
4.629
4.971
–0.008
4.290
justed to 665 mm (i.e. +25 mm) and its length was
the same as in experiments with ploughing on the
left side of the moving tractor. The obtained results
indicate that the ploughing operation on tractor’s
right side was subjected to the same rules as in a
series of measurements performed on the left side
of ploughing tractor.
This analysis indicates that the moment MTx increased with the increasing force working in the
upper drawbar; this means that the load of wheels
moving in the furrow is reduced. This is quite logical because the increasing tracking force in the upper drawbar always causes a deloading of plough’s
supporting wheel. The centre of gravity of a fourfurrow mounted plough used in these experiments
was situated on the left side of the tractor’s longitudinal axis. In case that the supporting wheel is
deloaded, the mass of the tractor influences more
the opposite driving wheels, which moves on the
field surface (i.e. the left one when ploughing on the
right side of the tractor and vice versa).
Ploughing on right; for abbreviations see Table 1
that with the increasing force working in the upper
drawbar the wheel moving in the furrow is de facto
deloaded (Fig. 2).
If the difference in forces influencing the left and
the right rear driving wheels (FLc – FPc) is denoted
as a value dependent on the moment MTx (Fig. 5),
then it is possible to say that the length of the upper
drawbar influences the load of wheels in the furrow
and on the field. As far as the pulling characteristics
of a tractor are concerned it is possible to conclude
that the length of the upper drawbar of the threepoint linkage influences the load of wheels moving
in the furrow and on the field surface. With regard
to the pulling characteristics of tractors this observation is significant.
The series of measurements presented in Table 3
refer to ploughing performed on the right side
of the moving tractor. The upper drawbar of the
three-point linkage was adjusted to two different
lengths. For the measurements marked in Table 3
as Groups A and B the upper drawbar was adjusted
to the same lengths as in experiments with left-side
ploughing, i.e. to 640 mm and 665 mm for groups
A and B, respectively. The measured data indicate
that in Group A with the shortest upper drawbar
marked tractive forces were registered in all cases
and that their average value was 11.22 kN. Group B
involves measurements performed on drawbars ad
Discussion
Renius (1987) mentioned that (under conditions
of 20% slippage) the ploughing aggregate consisting
of a tractor and a four-furrow reversible mounted
plough showed 45% and 25% of load on the wheel
moving in the furrow and on the field surface, respectively.
Results presented in this paper were obtained
also in experiments with attached four-furrow reversible plough. The average slippage was 15%.
Results of regression analysis of measured and calculated data are presented in Figs 4–7. The linear
dependence of the course of MTx moment on the
force working in the upper drawbar Fh of the threepoint linkage is illustrated in Figs 4 and 6. It can
be concluded that the greater the force working in
the upper bar, the lower the load of the rear supporting wheel and the higher the moment MTx. This
means that when ploughing on the left side of the
tractor the negative moment MTx increased with
the increasing value of the force Fh. As shown in
Fig. 2, the x-axis is oriented against the direction of
tractor’s movement and for that reason the negative moments operating along the x axis deload the
wheel moving in the furrow.
This is quite logical because when ploughing with
the deloaded supporting wheel of the plough the
moment MTx must be absorbed in lifting drawbars
85
Vol. 57, 2011, No. 3: 79–87
Res. Agr. Eng.
Fig. 6. Dependence of differences in forces
influencing the rear wheels FPc – FLc on MTx
momentum
FPc – FLc (kN)
Ploughing on right
MTx (kNm)
MTx (kNm)
Fig. 7. Dependence of MTx momentum on
force Fh in the upper drawbar
Fh (kN)
of the three-point linkage. The moment MTx then
influences the rear wheels and reduces the response
of soil to the wheel moving in the furrow. This is
documented in Figs 4 and 6, which illustrate the
dependence of differences in this reaction of soil to
moving wheels on the negative moment MTx.
It results from Tables 2 and 3 as well as from Figs 5
and 7 that with the increasing forces working in
the upper drawbar Fh the difference between the
load of wheel driving in the furrow and that moving on the field surface decreases. This fact shows a
positive effect on tractive characteristics of the rear
axle. On the other side, however, it is also possible
that with the increasing force occurring in the upper drawbar the front axle of the tractor might be
undesirably deloaded. To prevent this, it is always
necessary to add a sufficient weight to the front axle
of a tractor with an attached reversible plough.
86
Ploughing on right
References
Bauer F., Mišun V., Loprais A., 1989. Transformace
měřených silových účinků v táhlech tříbodového závěsu
do těžiště traktoru agregovaného s neseným pluhem
(Transformation of measured power effects in drawbars
of three-point linkage to tractor aggregated with mounted
plough). Zemědělská technika, 35: 205–214.
Chalupa M., Kratochvíl C., Kotek V., Heriban P., 2006.
Kombinovaná metoda analýzy dynamických vlastností
hnací soustavy (Combined method of analysis of driving
systems dynamic properties). In: Proceedings of Special
Technology 2006. Bratislava, STUB: 165–196.
Chalupa M., Veverka J., Vlach R., 2009. Influence of
design parameters on vehicle track dynamic loading. In:
Proceedings of the 2nd International Multi-Conference
on Engineering and Technological Innovation. Orlando,
IIIS: 365–369.
Res. Agr. Eng.
Kazzaz A., Grečenko A., 1968. Rozdíly mezi výkonností
kolového traktoru s pluhy a mezi údaji tahové zkoušky
(Difference between the power of wheel tractor with
ploughs and the haul test data). Zemědělská technika, 14:
131–152.
Mišun V., Bauer F., 1992. Zatížení hnacích kol traktoru při
orbě s nesenám pluhem (Load of powered wheels at ploughing with mounted plough). Zemědělská technika, 38: 1–9.
Mouazen A.M., Maleki M.R., Cockx L., Van Meirvenne
M., Van Holm L.H.J., Mercksx R., De Baerdemaker J.,
Ramon H., 2009. Optimum three-point linkage set up for
improving the quality of soil spectra and the accuracy of
soil phosphorus measured using an on-line visible and near
infrared sensor. Soil & Tillage Research, 103: 144–152.
Novák, P., Šmerda, T., Čupera, J., 2009. Impact of semitrailer tyre pressure upon passive resistance and tractor
Vol. 57, 2011, No. 3: 79–87
unit operation economy. Research in Agricultural Engineering, 55: 129–135.
Renius T., 1987. Traktoren Technik und ihre Anwendung.
München, BLV Verlagsgeselschaft: 191.
Sloboda A., Salanci J., Bugár T., 1997. Measuring and
evaluating of axial forces magnitudes in the lower drawbars
at the ploughing. Transactions of the Technical University
of Košice, 3: 1–12.
Šmerda T., Čupera J., 2010. Tire inflation and its influence
on drawbar characteristics and performance-energetic
indicators of a tractor set. Journal of Terramechanics, 47:
395–400.
Upadhyaya S.K., Kemble L.J., 1985. Accuracy of mounted
implement draft prediction using strain gages mounted
directly on three-point linkage system. Transactions of
the ASABE, 28: 40–60.
Received for publication June 11, 2010
Accepted after corrections January 19, 2011
Corresponding author:
Doc. Ing. Libor Severa, Ph.D., Mendel University in Brno, Faculty of Agronomy,
Zemědělská 1, 603 00 Brno, Czech Republic
phone: + 420 545132093, email: [email protected]
87